I don't read that as placing constraints on what a militia is. I read that as Congress can call them into service of the US (provide for organizing), arm them, discipline the militia and have rule (govern) over only those that are in the service of the US government. That does not define them nor does it provide for defining what a militia is.
I would disagree. The power to organize is the power to define. Regardless, that has nothing to do with 2nd A. rights, which belong directly to the People, not to the militia.